Concurrence of Stochastic 1-Qubit Maps

TL;DR

This paper derives explicit formulas for the concurrence of all stochastic 1-qubit maps, revealing a new convex roof pattern and demonstrating the existence of two-component optimal decompositions, with implications for higher-dimensional systems.

Contribution

It introduces a novel method to compute concurrence for stochastic 1-qubit maps and extends results to $2 imes n$ systems, providing new tools for entanglement quantification.

Findings

Explicit concurrence formulas for all stochastic 1-qubit maps

Existence of two-component optimal decompositions

Extension to $2 imes n$ systems and bounds on entanglement of formation

Abstract

Explicit expressions for the concurrence of all positive and trace-preserving ("stochastic") 1-qubit maps are presented. By a new method we find the relevant convex roof pattern. We conclude that two component optimal decompositions always exist. Our results can be transferred to $2 \times n$-quantum systems providing the concurrence for all rank two density operators as well as a lower bound for their entanglement of formation.